Brachistochrones are the shortest
François Rouvière
INTRODUCTION. The word brachistochrone (Greek for "shortest time") was created by Johann Bernoulli to qualify a problem he had solved, then submitted to the "most acute math- ematicians in the entire world" in the June 1696 issue ofActa Eruditorum :
given two points A and B, …nd the curve that a point, moving from A toB in a vertical plane under its own gravity, must follow so that, starting from A with zero velocity, it reaches B in the shortest possible time.
The question actually drew attention of some of the greatest minds of the time such as Leib- niz, l’Hospital, Newton and Johann’s brother Jakob Bernoulli. Taken up again and generalized in the eighteenth century by Euler and Lagrange, Bernoulli’s problem gave rise to the calculus of variations. More details on this fascinating chapter in the history of mathematical analysis can be found in Nahin [3] or Tikhomirov [4].
The brachistochrone problem is now a popular exercise for undergraduates, solved by ap- plication of the classical Euler-Lagrange di¤erential equations. The purpose of this note is to solve by more elementary tools (i.e. without any knowledge of the calculus of variations) a class of problems including Bernoulli’s and to establish the minimal character of the solution. The latter fact does not follow directly from the Euler-Lagrange equations, only giving a necessary condition for an extremum. We also include a short proof of a more general and slightly less elementary result (Theorem 3).
MAIN RESULT. Let : [t0; t1] ! R2, i.e. (t) = (x(t); y(t)), be a path in the Euclidean plane with continuous derivatives x0 = dx=dt, y0 = dy=dt, such that 0(t) 6= 0 for t0 < t < t1. The parameter t can then be replaced by the arclength, denoted by s. We call this a regular path. If f is a continuous function on an open set containing the image of , we consider the integral
F( ) :=
Z t1
t0
f(x(t); y(t))p
x0(t)2+y0(t)2 dt= Z
f(x; y) ds. Our main result deals with the special case of a function f of one variable only.
Theorem 1 Letf be a strictly positive continuous function on an intervalI inR. Let :I !R be a continuous function such that, for all y2I,
f(y) cos (y) =C , (1)
where C is a constant. Let (s) = (x(s); y(s)) ba a solution of the di¤ erential system
x0 = cos (y) , y0= sin (y) . (2)
Then gives the integral
F( ) = Z
f(y) ds
a minimal value among all regular paths inR I with the same endpoints.
Proof. Let (s) = (x(s); y(s)), s0 s s1 such that (2) holds, and let (t) = (X(t); Y(t)), t0 t t1, be another regular path inR I with the same endpoints as , i.e.
(t0) = (s0) = (x0; y0) , (t1) = (s1) = (x1; y1) . Let
A= Z x1
x0
C dx+ Z y1
y0
f(y) sin (y) dy .
On the one hand, changing variables by x = X(t), y = Y(t) we obtain from (1) and Cauchy- Schwarz inequality
A =
Z t1
t0
(CX0+f(Y)Y0sin (Y))dt
= Z t1
t0
f(Y)(X0cos (Y) +Y0sin (Y))dt Z t1
t0
f(Y)p
X02+Y02 dt=F( ).
On the other hand, changing variables byx=x(s),y=y(s) we obtain from (1) and (2)
A =
Z s1
s0
f(y)(x0cos (y) +y0sin (y))ds
= Z
f(y) ds=F( ) , whence the theorem.
Motivation. To motivate the solution (1)(2) of the problem, one might of course write down the corresponding Euler-Lagrange equations. But a more instructive motivation follows from an optical analogy observed by Johann Bernoulli himself. IndeedR
f(y)dsis, up to a constant factor, the travelling time of a light ray in a nonhomogeneous optical medium with refractive indexf(y)at the point (x; y); this occurs for instance when studying mirages. Discretizing the problem, Bernoulli replaced this continuous medium by a large number of horizontal layers with increasing heights y1; :::; yn and constant indices f(y1); :::; f(yn), and the path by a broken line. When crossing a layer the light ray obeys Snell’s law of refraction
f(yi) cos i=f(yi+1) cos i+1 ,
where i (resp. i+1) is the angle between the ray and the horizontal below (resp. above) the layer at height yi. Letting n go to in…nity Bernoulli "proved" in this way that, calling the angle between the path and the horizontal (whence the di¤erential equations (2)), f(y) cos must remain constant (whence (1)).
EXAMPLES. Theorem 1 applies to several classical problems.
a. Iff = 1it gives one more proof that the shortest path between two points in the Euclidean plane is the straight line.
b. Iff(x; y) = 2 y the integral F( ) is the area generated by rotating around thex-axis; the problem is to …nd minimal surfaces of revolution.
c. If f(x; y) = 1=y with y > 0, we obtain the geodesics of the Poincaré upper half-plane; the classical proof of their minimal property makes use of the transitive action of the isometry group SL(2;R).
d. The original brachistochrone problem corresponds to f(x; y) = 1=py with y > 0 ; the
e. An interesting variant of Bernoulli’s question is the problem of fast tunnels through the earth (Nahin [3] p. 229, Tung-Po Lin [5]):
given two points A and B on the surface of the earth (or inside), …nd the shape of a tunnel from A to B such that a point starting from A with zero velocity reaches B in the shortest possible time, by a frictionless motion under its own gravity; the earth is assumed to be a sphere of homogeneous density.
Solving the corresponding Euler-Lagrange equations one …nds hypocycloids ([5]). The problem can also be solved by the following polar coordinates variant of Theorem 1.
Corollary 2 Let f be a strictly positive continuous function on an interval I =]0; a[. Let :I !R be a continuous function such that, for all r2I,
rf(r) cos (r) =C , (1’)
whereCis a constant. In polar coordinates(r; )let (t) = (r(t); (t))be a regular path satisfying the di¤ erential equations
r0 =rsin (r) , 0 = cos (r) . (2’)
Then gives the integral
F( ) = Z
f(r) ds
a minimal value among all regular paths with the same endpoints.
Proof. Let = lnr. For any path F( ) =
Z t1
t0
f(r)p
r02+r2 02 dt= Z t1
t0
e f(e )
q 02+ 02 dt .
The result thus follows from Theorem 1 with( ; )and rf(r) instead of(x; y)and f(y).
Remark. By (2’) =2 is the angle at the pointM = (r; ) between the vectorOM! and the tangent to atM. In the optical analogy, Snell’s law (1) is now replaced by Bouguer’s formula (1’) for a spherically symmetric optical medium with refractive index f(r) (Born and Wolf [1]
p. 130). In the tunnel problem the functionf isf(r) = (a2 r2) 1=2.
GENERALIZATION.A crucial fact in the proof of Theorem 1 is of course that the di¤erential form
!=f(y)(cos (y) dx+ sin (y) dy)
is closed if (and only if)f(y) cos (y)is constant. The method extends, under stronger regularity assumptions, by means of "Hilbert’s independence integral" (Young [6] p. 27, Born and Wolf [1] Appendix 1). It leads to the following less elementary result.
LetT be an open interval and U; V open subsets ofRnwithU simply connected. We denote by L : (t; x; v) 7 ! L(t; x; v) a real-valued C2 function (the Lagrangian) on T U V and by p: (t; x)7!p(t; x) a C1 map fromT U intoV. Let! be the di¤erential form
!:= (L(t; x; p) @vL(t; x; p) p) dt+@vL(t; x; p) dx onT U, with p=p(t; x).
In Theorem 1 we had p(t; x1; x2) = (cos (x2);sin (x2)), L(t; x; v) = f(x2)kvk (Euclidean norm), T = R, U = R I, V = R2; the convexity argument in (ii) below was replaced by Cauchy-Schwarz inequality.
Theorem 3 Assume ! is closed in T U.
Let be any solution of the di¤ erential system 0(t) =p(t; (t)), C1 on an interval [a; b] T, such that (t)2U and 0(t)2V for all t2[a; b].
(i) Then satis…es the Euler-Lagrange system of di¤ erential equations d
dt(@vL(t; ; 0)) =@xL(t; ; 0) . Thusp is a "…eld of extremals".
(ii) Assume furthermore that V is convex and v7! L(t; x; v) is a convex function on V for each (t; x)2T U. Then
F( ) :=
Z b
a L(t; (t); 0(t)) dt F( )
for allC1 paths : [a; b]!U with the same endpoints as and such that 0(t)2V for all t.
Proof. (i) Let h: [a; b]!Rn be anyC1 path withh(a) =h(b) = 0 and let, for"2R,
"(t) := (t) +"h(t)
be a variation of . If the constant "is small enough we have (t; "(t); 0"(t))2 T U V for allt and
0" p(t; ") =p(t; ) p(t; ") +"h0=O("),
uniformly with respect to t, whence
L(t; "; 0") =L(t; "; p(t; ")) +@vL(t; "; p(t; ")) ( 0" p(t; ")) +O("2)
and, by integration,
F( ") =
Z
"
!+O("2) .
Since ! is closed and " has the same endpoints as , the integral R
"! is independent of ".
Therefore the derivative d"dF( ")vanishes at "= 0, and the Euler-Lagrange equations follow by a classical computation.
(ii) By convexity ofLwe have
L(t; ; 0) L(t; ; p) +@vL(t; ; p) ( 0 p) with = (t), 0 = 0(t),p=p(t; (t))and, by integration,
F( ) Z
!= Z
! =F( );
in the latter equalities we successively used the assumptions that ! is closed, and have common endpoints and 0 =p(t; ).
Remark. Taking = +"hin(ii) would imply the Euler-Lagrange equations again, since F( ) is a minimum.
REFERENCES
1. M. Born and E. Wolf, Principles of optics, 7th edition, Cambridge University Press, 1999.
2. R. Courant,Calculus of variations, Courant Institute, 1962.
4. V. M. Tikhomirov, Stories about maxima and minima, Math. World vol. 1, Amer. Math.
Soc., 1990.
5. Tung-Po Lin, Fast tunnels through the earth, Amer. Math. Monthly, June-July 1969, 708- 709.
6. L. C. Young, Calculus of variations and optimal control theory, W. B. Saunders Company, 1969.
Laboratoire J. A. Dieudonné, Université de Nice Parc Valrose, 06108 Nice cedex 2, France
e-mail: [email protected]
